/Subtype/Type1 27 0 obj /Subtype/Type1 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 826.4 531.3 958.7 1076.8 Frobenius’ method for curved cracks 63 At the same time the unknowns B i must satisfy the compatibility equations (2.8), which, after linearization, become 1 0 B i dξ=0. << \end{equation*}, In the following, the zeros $\lambda _ { i }$ of the indicial polynomial will be ordered by requiring, \begin{equation*} \operatorname { Re } \lambda _ { 1 } \geq \ldots \geq \operatorname { Re } \lambda _ { \nu }. /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 n: 2. 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 441.3 461.2 353.6 557.3 473.4 (3.6) 4. SINGULAR POINTS AND THE METHOD OF FROBENIUS 291 AseachlinearcombinationofJp(x)andJ−p(x)isasolutiontoBessel’sequationoforderp,thenas wetakethelimitaspgoeston,Yn(x)isasolutiontoBessel’sequationofordern.Italsoturnsout thatYn(x)andJn(x)arelinearlyindependent.Thereforewhennisaninteger,wehavethegeneral In the Frobenius method one examines whether the equation (2) allows a series solution of the form. Note that neither of the special cases below does exclude the simple generic case above. You were also shown how to integrate the equation to … A similar method of solution can be used for matrix equations of the first order, too. An adaption of the Frobenius method to non-linear problems is restricted to exceptional cases. 652.8 598 0 0 757.6 622.8 552.8 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 384.3 611.1 675.9 351.8 384.3 643.5 351.8 1000 675.9 611.1 675.9 643.5 481.5 488 /FirstChar 33 /LastChar 196 /BaseFont/KNRCDC+CMMI12 531.3 826.4 826.4 826.4 826.4 0 0 826.4 826.4 826.4 1062.5 531.3 531.3 826.4 826.4 /FirstChar 33 The method looks simpler in the most common case of a differential operator, \tag{a9} L = a ^ { [ 2 ] } ( z ) z ^ { 2 } \left( \frac { d } { d z } \right) ^ { 2 } + a ^ { [ 1 ] } ( z ) z \left( \frac { d } { d z } \right) + a ^ { [ 0 ] } ( z ). << 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 665 570.8 924.4 812.6 568.1 670.2 15 0 obj If r1¡r2= 0, the solution basis of the ODE(1)is given by y1(x) =xr1. For any $i = 1 , \dots , \nu$, the zero $\lambda _ { i }$ of the indicial polynomial has multiplicity $n _ { i } \geq 1$, but none of the numbers $\lambda _ { 1 } - \lambda _ { i } , \ldots , \lambda _ { i - 1 } - \lambda _ { i }$ is a natural number. The method of Frobenius gives a series solution of the form y(x) = X∞ n=0 an (x −c)n+s where p or q are singular at x = c. Method does not always give the general solution, the ν = 0 case of Bessel’s equation is an example where it doesn’t. 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1. In particular, this can happen if the coe cients P(x) and Q(x) in the ODE y00+ P(x)y0+ Q(x)y = 0 fail to be de ned at a point x 0. /Type/Font endobj /Widths[791.7 583.3 583.3 638.9 638.9 638.9 638.9 805.6 805.6 805.6 805.6 1277.8 /Type/Font 0 is y(x) … cxe1=x, which could not be captured by a Frobenius expansion. 33 0 obj In this case the leading behavior of y(x) as x ! Putting $\lambda = \lambda _ { i }$ in (a6), obtaining solutions of (a3) can be impossible because of poles of the coefficients $c_j ( \lambda )$. All the three cases (Values of 'r' ) are covered in it. /Type/Font Commonly, the expansion point can be taken as, resulting in the Maclaurin series (1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 458.3 458.3 416.7 416.7 /FirstChar 33 /LastChar 196 www.springer.com The approach does produce special separatrix-type solutions for the Emden–Fowler equation, where the non-linear term contains only powers. 687.5 312.5 581 312.5 562.5 312.5 312.5 546.9 625 500 625 513.3 343.8 562.5 625 312.5 Here, $\epsilon > 0$, and for an equation in normal form, actually $\epsilon \geq r$. 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] 638.4 756.7 726.9 376.9 513.4 751.9 613.4 876.9 726.9 750 663.4 750 713.4 550 700 720.1 807.4 730.7 1264.5 869.1 841.6 743.3 867.7 906.9 643.4 586.3 662.8 656.2 1054.6 /Subtype/Type1 /FontDescriptor 26 0 R 481.5 675.9 643.5 870.4 643.5 643.5 546.3 611.1 1222.2 611.1 611.1 611.1 0 0 0 0 /LastChar 196 FROBENIUS SERIES SOLUTIONS 5 or a n = a n 1 5n+ 5r+ 1; n= 1;2;:::: (35) Finally, we can use the concrete values r= 1 and r= 1 5. Method of Frobenius Example First Solution Second Solution (Fails) What is the Method of Frobenius? /Widths[1062.5 531.3 531.3 1062.5 1062.5 1062.5 826.4 1062.5 1062.5 649.3 649.3 1062.5 Computation of the polynomials $p _ { j } (\lambda)$. Let $\mathbf{N}$ denote the set of natural numbers starting at $1$ (i.e., excluding $0$). The other solution takes the form y2(t) = y1(t)lnt + tγ1 + 1 ∞ ∑ n = 0dntn. Let y=Ún=0 ¥a xn+r. /BaseFont/XKICMY+CMSY10 666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 /BaseFont/IMGAIM+CMR8 /Name/F3 3. 351.8 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 351.8 351.8 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 EnMath B, ESE 319-01, Spring 2015 Lecture 4: Frobenius Step-by-Step Jan. 23, 2015 I expect you to 2. In this video, I introduce the Frobenius Method to solving ODEs and do a short example.Questions? 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 Frobenius Method ( All three Cases ) - Free download as PDF File (.pdf), Text File (.txt) or read online for free. /Name/F4 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 << /FontDescriptor 35 0 R \end{equation*}. Since a change x-x 0 ↦ x of variable brings to the case that the singular point is the origin, we may suppose such a starting situation. The next two theorems will enable us to develop systematic methods for finding Frobenius solutions of ( eq:7.5.2 ). One gets $L _ { 0 } ( u ^ { \lambda } ) = \pi ( \lambda ) z ^ { \lambda }$ with the indicial polynomial, \tag{a5} \pi ( \lambda ) = \sum _ { n = 0 } ^ { N } ( \lambda + n ) ( \lambda + n - 1 ) \ldots ( \lambda + 1 ) a ^ { n _0} = , \begin{equation*} = a _ { 0 } ^ { N } \prod _ { i = 1 } ^ { \nu } ( \lambda - \lambda _ { i } ) ^ { n _ { i } }. /FontDescriptor 17 0 R 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 590.3 767.4 795.8 795.8 1091 Suppose $\lambda _ { 1 } - \lambda _ { 2 } \in \mathbf{N}$. 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 { l ! } /LastChar 196 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 There is at least one Frobenius solution, in each case. all with $\lambda = \lambda _ { 2 }$ and $l = 0 , \dots , n _ { 2 } - 1$, are $n_{2}$ linearly independent solutions of the differential equation (a3). Because for $i = 1 , \dots , \nu$ and $l = 0 , \dots , n _ { i } - 1$, all leading terms are different, the method of Frobenius does indeed yield a fundamental system of $N$ linearly independent solutions of the differential equation (a3). This is a method that uses the series solution for a differential equation, where we assume the solution takes the form of a series. 2n 2, so Frobenius’ method fails. >> << /FontDescriptor 8 0 R endobj 12 0 obj 30 0 obj 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 stream The leading term $b _ { l0 } ( \operatorname { log } z ) ^ { l } z ^ { \lambda _ { i } }$ is useful as a marker for the different solutions. The indicial polynomial is simply, \begin{equation*} \pi ( \lambda ) = ( \lambda + 2 ) ( \lambda + 1 ) a ^ { 2_0 } + ( \lambda + 1 ) a ^ { 1_0 } + a ^ { 0_0 } = \end{equation*}, \begin{equation*} = a ^ { 2 } o ( \lambda - \lambda _ { 1 } ) ( \lambda - \lambda _ { 2 } ). In the following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand Georg Frobenius. with $\lambda = \lambda _ { 2 }$ in the second function, are two linearly independent solutions of the differential equation (a9). >> /BaseFont/BPIREE+CMR6 /Widths[660.7 490.6 632.1 882.1 544.1 388.9 692.4 1062.5 1062.5 1062.5 1062.5 295.1 /Name/F5 • Back to Frobenius method for second solutions in three cases –n = = 0, the double root – Integer = n 0, roots differ by an integer, J-n(x) = (-1)nJ n(x) – Non-integer , easiest case, J and J- are two linearly independent solutions • General case for second solution [0,1] 2( ln() m m n endobj 21 0 obj 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 {\displaystyle u' (z)=\sum _ {k=0}^ {\infty } (k+r)A_ {k}z^ {k+r-1}} /Name/F2 /Subtype/Type1 Indeed (a1) and (a2) imply, \begin{equation*} L ( u ( z , \lambda ) ) = \end{equation*}, \begin{equation*} = [ \sum _ { i = 0 } ^ { \infty } \sum _ { n = 0 } ^ { N } a _ { i } ^ { n } z ^ { n + i } ( \frac { \partial } { \partial z } ) ^ { n } ] [ \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { \lambda + k } ] = \end{equation*}, \begin{equation*} = \sum _ { i = 0 } ^ { \infty } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { i } \sum _ { n = 0 } ^ { N } a _ { i } ^ { n } z ^ { n } \left( \frac { \partial } { \partial z } \right) ^ { n } z ^ { \lambda + k } = \end{equation*}, \begin{equation*} = \sum _ { i = 0 } ^ { \infty } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { i } p _ { i } ( \lambda + k ) z ^ { \lambda + k } = \end{equation*}, \begin{equation*} = z ^ { \lambda } \sum _ { j = 0 } ^ { \infty } z ^ { j } \left[ \sum _ { i + k = j } c _ { k } ( \lambda ) p _ { i } ( \lambda + k ) \right] = \end{equation*}, \begin{equation*} = c _ { 0 } z ^ { \lambda } \pi ( \lambda ) + \end{equation*}, \begin{equation*} + z ^ { \lambda } \sum _ { j = 1 } ^ { \infty } z ^ { j } \left[ c _ { j } ( \lambda ) \pi ( \lambda + j ) + \sum _ { k = 0 } ^ { j - 1 } c _ { k } ( \lambda ) p _ { j - k } ( \lambda + k ) \right]. /LastChar 196 u ( z ) = z r ∑ k = 0 ∞ A k z k , ( A 0 ≠ 0 ) {\displaystyle u (z)=z^ {r}\sum _ {k=0}^ {\infty }A_ {k}z^ {k},\qquad (A_ {0}\neq 0)} Differentiating: u ′ ( z ) = ∑ k = 0 ∞ ( k + r ) A k z k + r − 1. /Type/Font /Type/Font This is the extensive document regarding the Frobenius Method. << /FirstChar 33 ( \operatorname { log } z ) ^ { l } z ^ { \lambda _ { 2 } } + \ldots, \end{equation*}. 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 888.9 888.9 888.9 343.8 593.8 312.5 937.5 625 562.5 625 593.8 459.5 443.8 437.5 625 593.8 812.5 593.8 The European Mathematical Society. 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] /FirstChar 33 Let $1 \leq j \leq \nu$ and let $\lambda _ { i }$ be a zero of the indicial polynomial of multiplicity $n_i$ for $i = 1 , \dots , j - 1$. ���ů�f4[rI�[��l�rC\�7 ����Kn���&��͇�u����#V�Z*NT�&�����m�º��Wx�9�������U]�Z��l�۲.��u���7(���"Z�^d�MwK=�!2��jQ&3I�pݔ��HXE�͖��. 38 0 obj \end{equation*}, \begin{equation*} ( \frac { \partial } { \partial \lambda } ) ^ { m _ { j } + l } \left[ u ( z , \lambda ) ( \lambda - \lambda _ { j } ) ^ { m _ { j } } \right] = \end{equation*}, \begin{equation*} = \frac { ( m _ { j } + l ) ! } In the former case there’s obviously only one Frobenius solution. /FontDescriptor 32 0 R This fact is the basis for the method of Frobenius. The easy generic case occurs if the indicial polynomial has only simple zeros and their differences $\lambda _ { i } - \lambda _ { j }$ are never integer valued. Computation of the polynomials $p _ { j } ( \lambda )$. >> The functions, \begin{equation*} ( \frac { \partial } { \partial \lambda } ) [ u ( z , \lambda ) ( \lambda - \lambda _ { 2 } ) ] = z ^ { \lambda_2 } + \ldots , \end{equation*}. The Frobenius method has been used very successfully to develop a theory of analytic differential equations, especially for the equations of Fuchsian type, where all singular points assumed to be regular (cf. We classify a point x 2 Frobenius Series Solution of Ordinary Diﬀerential Equations At the start of the diﬀerential equation section of the 1B21 course last year, you met the linear ﬁrst-order separable equation dy dx = αy , (2.1) where α is a constant. 492.9 510.4 505.6 612.3 361.7 429.7 553.2 317.1 939.8 644.7 513.5 534.8 474.4 479.5 with $l = 0 , \dots , n _ { j } - 1$ and $\lambda = \lambda _ { j }$, are $n_j$ linearly independent solutions of the differential equation (a3). 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 826.4 295.1 826.4 531.3 826.4 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 0 0 707.2 571.2 544 544 816 816 272 1. \begin{equation*} ( \frac { \partial } { \partial \lambda } ) ^ { n _ { 1 } + l } [ u ( z , \lambda ) ( \lambda - \lambda _ { 2 } ) ^ { n _ { 1 } } ] = \end{equation*}, \begin{equation*} = \frac { ( n _ { 1 } + l ) ! } The point $z = 0$ is called a regular singular point of $L$. 624.1 928.7 753.7 1090.7 896.3 935.2 818.5 935.2 883.3 675.9 870.4 896.3 896.3 1220.4 1062.5 826.4] 708.3 795.8 767.4 826.4 767.4 826.4 0 0 767.4 619.8 590.3 590.3 885.4 885.4 295.1 endobj View Notes - Lecture 5 - Frobenius Step by Step from ESE 319 at Washington University in St. Louis. 805.5 896.3 870.4 935.2 870.4 935.2 0 0 870.4 736.1 703.7 703.7 1055.5 1055.5 351.8 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 Method of Frobenius. /Widths[609.7 458.2 577.1 808.9 505 354.2 641.4 979.2 979.2 979.2 979.2 272 272 489.6 /Widths[351.8 611.1 1000 611.1 1000 935.2 351.8 481.5 481.5 611.1 935.2 351.8 416.7 \begin{equation*} u ( z , \lambda _ { i } ) = z ^ { \lambda _ { i } } + \ldots , \end{equation*}, \begin{equation*} \frac { \partial } { \partial \lambda } u ( z , \lambda _ { i } ) = ( \operatorname { log } z ) z ^ { \lambda_i } +\dots \dots \end{equation*}, \begin{equation*} \left( \frac { \partial } { \partial \lambda } \right) ^ { ( n _ { i } - 1 ) } u ( z , \lambda _ { i } ) = ( \operatorname { log } z ) ^ { n _ { i } - 1 } z ^ { \lambda _ { i } } +\dots \end{equation*}. /BaseFont/TBNXTN+CMTI12 /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 Complications can arise if the generic assumption made above is not satisfied. 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 (You should check that zero is really a regular singular point.) << >> /FirstChar 33 There is a theorem dealing However, the method of Frobenius can be extended to the case where , , and are functions that can be represented by power series in on some interval that contains zero, and . 531.3 531.3 413.2 413.2 295.1 531.3 531.3 649.3 531.3 295.1 885.4 795.8 885.4 443.6 The method of Frobenius works for differential equations of the form y00 +P(x)y0 +Q(x)y=0 in which P or Q is not analytic at the point of expansion x 0. 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